192 XIII INTEGRATION By writing each /*,- as a linear combination of four positive measures (13.3.6), we may assume that the measures \JLJ are all positive. Then we take r ^ == X Af/> and apply the result of (13.15.1) to each \JLJ g L (13.15.3) (i) With respect to the order relation on MR(X), any two real measures \i, v have a least upper bound sup(/j, v) and a greatest lower bound inf(/*, v). For each real measure \JL put /z+ = sup^u, 0) and /i~ = sup( /*, 0); then we have (13.15.3.1) inf(^+, iT) = 0, /* = and, for any two real measures /i, v, v = sup(/z, v) + inf(^, v), (13.15.3.2) i, v) = i(/^ + v + b vl)> inf(//, v) = i(/( + v |/i v|). (ii) Le^ 1 be a positive measure and let g^ g2 be real-valued locally k-integrable functions on X. If fa = #1 * A <2/?Ģ/ ^J2 = ^2 / m particular, for any real-valued locally X-integrable function g, (13.15.3.4) (g A)+ =g+ - ^ (g A)' = <T ' A. PFe /?m;e ^ A ^ g2 ' A if and only if gą(x) ^ g2(x) almost everywhere with respect to L (iii) Let (gn) be an increasing sequence of locally A-integrable real-valued functions. Then the increasing sequence of measures gn 1 is bounded above in MR(X) if and only if the function sup gn is locally k-integrable, and in that case we have n (13.15.3.5) supte.-A To prove (i), we remark that the measures n, v can be written in the form ^ = ^-^2, v = v1-v2, where /^, /*2, Y! and v2 are positive (13.3.6). Applying (13.15.2) to the four measures jjLl9 /i2, v1? v2 we see (by using (13.13.2)) that /i = u - 1 and v = i? A, where A is a positive measure and u, v m r